Simplify the following expression and state the condition under which the simplification is valid. $n = \dfrac{6z^3 - 30z^2 - 300z}{z^3 - 13z^2 + 30z}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ n = \dfrac {6z(z^2 - 5z - 50)} {z(z^2 - 13z + 30)} $ $ n = \dfrac{6z}{z} \cdot \dfrac{z^2 - 5z - 50}{z^2 - 13z + 30} $ Simplify: $ n = 6 \cdot \dfrac{z^2 - 5z - 50}{z^2 - 13z + 30}$ Since we are dividing by $z$ , we must remember that $z \neq 0$ Next factor the numerator and denominator. $ n = 6 \cdot \dfrac{(z - 10)(z + 5)}{(z - 10)(z - 3)}$ Assuming $z \neq 10$ , we can cancel the $z - 10$ $ n = 6 \cdot \dfrac{z + 5}{z - 3}$ Therefore: $ n = \dfrac{ 6(z + 5)}{ z - 3 }$, $z \neq 10$, $z \neq 0$